Given a function K(⋅, ⋅) that satisfies the hypotheses of Mercer’s theorem (and hence is a kernel function), one can use this function in the dual formulation of the classifier in Equation (7.49) to implicitly use the possibly infinite-dimensional transformed feature space Φ without needing to explicitly compute the corresponding feature vectors {ϕj}. In other words: the kernel function K induces the feature space Φ.

Note, again, that even though the feature vector ϕ(m) may have a very high, possibly even infinite dimensionality d∗, the classifier in Equation (7.49) is still fully determined by only N free parameters.

Examples of common kernel functions

As the first example, consider the trivial kernel

$K\left(m,u\right):=\langle m,u\rangle ={m}^{T}u.\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}$

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